Does MOSFET let current flow through source to drain as it allows it from drain to source?

Yes it does conduct in either direction.

Due to the body diode, most discrete MOSFETs cannot block in the reverse direction, but the channel will conduct in either direction when the gate is biased "on".

If you want to conduct and block in both directions you need two MOSFETs in series.

MOSFETs used as near-perfect rectifiers are usually used in the reverse direction for conduction (so they can block in the other direction).

Edit: Your schematic here:

Illustrates one example of switching AC with two MOSFETs (one of which will be conducting in reverse at any given time when the switches are on).

Another example is here, from the LT4351 datasheet:

In case anyone else stumbles across this thread, the TLP175A (and plenty of others) is a single component which directly implements the circuit suggested in the above answers. That particular IC features optocoupler isolation and is even suggested use as a relay replacement.